Journal of Applied Mathematics

Volume 2014 (2014), Article ID 450289, 15 pages

http://dx.doi.org/10.1155/2014/450289

Research Article

Stochastic Viscoelastic Wave Equations with Nonlinear Damping and Source Terms

Shuilin Cheng, Yantao Guo, and Yanbin Tang

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

Received 23 December 2013; Revised 11 February 2014; Accepted 11 February 2014; Published 27 March 2014

Academic Editor: Francisco J. Marcellán

Copyright © 2014 Shuilin Cheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The goal of this paper is to study an initial boundary value problem of stochastic viscoelastic wave equation with nonlinear damping and source terms. Under certain conditions on the initial data: the relaxation function, the indices of nonlinear damping, and source terms and the random force, we prove the local existence and uniqueness of solution by the Galerkin approximation method. Then, considering the relationship between the indices of nonlinear damping and nonlinear source, we give the necessary conditions of global existence and explosion in finite time in some sense of solutions, respectively.

1. Introduction

We consider a stochastic viscoelastic wave equation with nonlinear damping and source terms

where is a bounded domain in with smooth boundary , , , is a given positive constant which measures the strength of noise, and is an infinite dimensional Wiener process, is -valued progressively measurable and is a positive relaxation function satisfying some conditions to be specified later. By simplicity, we have set equal to 1 all the coefficients in the equation different from the random force.

For the deterministic case on viscoelastic wave equation, many authors studied the following problem:

where . If , . Haraux and Zuazua [1] and Kopáčková [2] proved that the damping term assures existence of global solution and decay of solution for arbitrary initial data. If , , Ball [3] and Kalantarov and Ladyzhenskaya [4] gave that the source term causes finite time blow-up with the large initial data. If , , the interaction between damping term and source term occurs; Levine et al. [5, 6] studied the linear damping (i.e., ) and proved that the solution with negative initial energy blows up in finite time; Georgiev and Todorova [7] considered nonlinear damping and source terms; they showed that the solution blows up in finite time if for sufficiently large initial data and exists globally if with large initial data. Alves et al. [8] and Rammaha [9] focused the nonlinear wave equations or systems on the influence between damping and source and described the existence, uniform decay rates, and blow-up to the solutions.

In fact, lots of investigators have paid attention to the viscoelastic wave equation, which has its origin in the mathematical description of viscoelastic materials. The dynamic properties of viscoelastic materials are of great importance as they appear in many applications to natural sciences. The general viscoelastic wave equation has the following form:

where is a relaxation function and , are given functions. If is nonlinear, is linear; Kafini and Messaoudi [10] established a blow-up result; if , . Messaoudi [11] showed that the solution with negative initial energy blows up in finite time if and exists globally if under suitable conditions on relaxation function . This blow-up result has been pushed to the case of positive initial energy by Messaoudi [12]. For , Song and Zhong [13] obtained that the solution with positive initial energy blows up in finite time. Ikehata [14] gave some remarks on the wave equations with nonlinear damping and source terms. Aassila et al. [15], Cavalcanti et al. [16], and Cavalcanti et al. [17] studied the boundary damping and proved the existence and uniform decay of the solutions. Cavalcanti et al. [18] discussed the asymptotic stability of the wave equation on a compact Riemannian manifold; they proved that the solutions of the corresponding partial viscoelastic model decay exponentially to zero under some conditions.

Under the consideration of random environment, some authors investigated the following stochastic wave equation with nonlinear damping and source terms:

where is a -valued -Wiener process on some completed probability space, and is a nonnegative operator with finite trace on (see [19–26]). If , or ; Bo et al. [27] showed that the solution blows up with positive probability or it is explosive in sense. If , Gao et al. [28] showed that the global solution exists for , and the solution blows up with positive probability or is explosive in energy sense for .

Recently, Wei and Jiang [29] and Liang and Gao [30] considered the following nonlinear stochastic viscoelastic wave equation with linear damping:

and they used the fixed point theorem to prove the existence and uniqueness of local mild solution; then by an appropriate energy inequality and estimations, they obtained the global existence and the decay estimate of the energy function of the solution and showed that the solution blows up with positive probability or it is explosive in sense under some conditions.

As we know, no one considers the stochastic viscoelastic wave equation (1) with the interaction between nonlinear damping and nonlinear source terms. In this paper, we study the global existence and the explosive phenomena under some suitable conditions on the nonlinear damping and nonlinear source terms.

In contrast with the model in [27], we add a viscoelastic term and use the nonlinear damping term instead of the linear damping and the strong damping . To the model in [28], we add a viscoelastic term . To the model in [29, 30], we use the nonlinear damping term instead of the linear damping . To the model in [12], we add a random force. In this paper, we generalize the blow-up and global existence results to the solution of (1) with interaction among viscoelastic memory, nonlinear damping, nonlinear source, and random force.

This paper is organized as follows. In the next section, we recall some preliminaries related to assumptions and definitions for the solutions of the stochastic equations. In Section 3, we use the Galerkin approximation method to get the local solution of stochastic viscoelastic wave equations with nonlinear damping and source terms. In Section 4, by the energy function and some estimates, we prove that the solution blows up with positive probability or it is explosive in energy sense for . In the last section, we obtain the existence of global solution by the Borel-Cantelli Lemma.

2. Preliminaries

Let be a separable Hilbert space with Borel -algebra , and let be a probability space. We set with the inner product and norm denoted by and , respectively. Denote by the norm for and by the Dirichlet norm in which is equivalent to norm. We also assume that , satisfy

Lemma 1 (see [27]). For all and or , there exists a constant such that

One assumes that is a bounded nonincreasing function satisfying , , and there exist positive constants and such that

In this paper, stands for expectation with respect to probability measure , is a -valued -Wiener process on the probability space with the covariance operator satisfying . A complete orthonormal system in with , and a bounded sequence of nonnegative real numbers satisfies that , .

To simplify the computations, we assume that the covariance operator and Laplacian with homogeneous Dirichlet boundary condition have a common set of eigenfunctions, that is,

and then, for any , has an expansion

where are real valued Brownian motions mutually independent on . Let be the set of -valued processes with the norm

where denotes the adjoint operator of . For any process , we can define the stochastic integral with respect to the -Wiener process as , which is a martingale. For more details about the infinite dimension Wiener process and the stochastic integral, we refer the readers to [21].

Definition 2. Assume that , and ; is said to be a solution of (1) on the interval , if is -valued progressively measurable, , , and such that (1) holds in the sense of distributions over for almost all .

3. Local Existence and Uniqueness

In this section, we establish the local existence and uniqueness of solution to problem (1) by the Galerkin approximation method. Set , . For each , we define a cut-off function , such that , for , and

Denote for , then, Lemma 1 implies that

where is a constant depending only on .

For any , the Yosida approximation of mapping is

and it has the following properties (see [28, 31, 32]):

Lemma 3 (see [28]). Let be a sequence of positive numbers, and let be a sequence of real numbers such that and as , then

Lemma 4 (see [33]). Let be a bounded domain in . Suppose that is a bounded sequence in , such that for almost all , for some . Then weakly in .

Fix and ; we consider the regularized initial boundary value problem

with the initial data

and is -valued progressively measurable such that

For notational convenience, we omit in the Hilbert space.

Lemma 5. Assume (18), (19), and the conditions on hold. Then there is a pathwise unique solution of (17) such that , and   ;  . Moreover, it holds that

where denotes a positive constant independent of .

Proof. Let and let be the solution of the following system:

By formula, we have

for all and almost all , where

Using Hölder’s inequality, Young’s inequality, Poincaré’s inequality, and Lemma 1, we have

and so we have

Since

thus we have

where depends on the fixed number .

Due to integration by parts, we have

Since is the Yosida approximation of mapping , by the Lemma 3 and sign-preserving theorem of limit, for small enough , we have

Moreover, by the conditions of , we get

Next, from the properties of , we have

and similar to the derivation of (34), we have

By the B-D-G inequality and Young’s inequality, we have

From (22)–(37), we get

then,

Set

and we can rewrite (39) as follows:

and due to Gronwall’s inequality and (29), it is clear that

Let be the orthogonal projection of into the space Span , such that

Define , where is an -valued progressively measurable such that (19) holds and is an -valued process; there is a subset with such that for each , , and we have

for all .

From (42), there is a subsequence , for each , such that,

and by the properties of relaxation function and Hölder’s inequality, we get

From (15) and embedding theorem we have

Together with (42)–(47), we obtain

for all . By (48) and weakly star in , we have

This implies that there exists a subsequence still denoted by such that

Due to (47) and Lemma 4, it is clear that

Thus, satisfies (17) in the sense of distributions over .

Next, we will prove the uniqueness of the solution. If there is another solution of (17), , in the above sense, then satisfies

Taking the inner product of (51) with in , we obtain

From (15), we have

By Lemma 1 and Hlder’s inequality,

Due to (30), we have

Combining (52) with (55), similar to (32) and (34), we get

which implies , that is, . So is well defined, for each .

Finally, we state that is -valued progressively measurable for any , and the energy inequality holds true; this can be established by the similar argument in [28, 31].

Moreover, we still fix and consider the following problem:

The following lemma is important to prove the local existence of solution of (1).

Lemma 6 (see [28, 31]). Assume that (18), (19), and the conditions of hold. Then there is a pathwise unique solution of (58) such that

From Lemmas 5 and 6, we state a local existence theorem of (1); the proof is standard; for more information we refer the readers to [28, 31].

Theorem 7 (see [28, 31]). Assume that , , (6), and the conditions of hold; there is a pathwise unique local solution of (1) according to Definition 2 such that the following energy equation holds:

4. Blow Up

In this section we prove our main result for . For this purpose, we give refined restrictions on and relaxation function such that

Define an energy function

where

For each , introduce the stopping time by , where is increasing in , let .

In order to prove our blow-up result, we rewrite (1) as an equivalent system

where . Then the energy function becomes

First we give a lemma.

Lemma 8. Assume that (6), (61), and the conditions of hold. Let be a solution of system (64) with initial data . Then we have

Proof. Using formula to and , respectively, and taking the expectations, in the same way as our discussions in existence of solution to deal with the memory term, it is easy to get (66) and (67) (see [28]).

Let

Due to (61), we have

We set . Then, (66) implies that

Lemma 9. Let be a solution of system (64). Then there exists a positive constant such that

Proof. If then by Sobolev embedding theorem. If then . Therefore, combination with the definition of energy function, we can get (71).

Theorem 10. Assume that (6), (61), and the conditions of hold. Let be a solution of system (64) with initial data satisfying

where is an arbitrary constant. If , then the solution and the lifespan defined above, either

(1), that is, blows up in finite time with positive probability, or(2)there exists a positive time such that

where

and are given in later.

Proof. For the lifespan of the solution of (1) with norm, firstly we treat the case when . Then, for sufficiently large , by (70) and (72), we have

Define , where

and is a very small constant determined in later.

Using (67) and (70), we obtain

and by the Hlder’s inequality,

Inserting (78) and (79) into (77), we get

From , by and Hlder’s inequality, we obtain the estimate of the last term in (80)

and the Young’s inequality implies that

where is a constant determined in later.

In view of (75), we have

where . We assume , (83), and (76) imply that

Combining (82) with (84), we arrive that

where .

Hence, substituting (85) into (80),

By Lemma 9 with and (86), we have

where .

Note that

and denote

We write , where , the estimate (87) yields

From (72) and (75), we obtain

Substituting (91) into (90), we get

Next, we can choose large enough so that (92) becomes

where is the minimum of the coefficients of , , , , and in (93). Once is fixed, we pick small enough so that

Therefore, (93) takes the form

Consequently we have

Since

it implies that

for .

We choose ; then , by (76) and (98) becomes

Using Lemma 9 with , we obtain

for all .

Therefore we have

for all . Combining (95) and (101),

where is a positive constant depending only on and ; then it yields

Let

Then as . This means that there exists a positive time such that

As for the case when (i.e., then blows up in finite time with positive probability.

The proof of Theorem 10 is completed.

Remark 11. (1) In the deterministic case of , it is well known that for , the condition and already imply that the solution blows up in finite time (see, e.g., [11]). In the stochastic case of , to balance the influence of such that the local solution of (1) blows up with positive probability or is explosive in sense, the initial energy and relaxation function should be satisfied that , and .

(2) Our results have included the case which is without viscoelastic term (i.e., satisfied (61)).

5. Global Existence

In this section we show that solution of (1) is global if . We use the Borel-Cantelli Lemma to prove the existence of global solution. For this aim, we introduce an energy function

Theorem 12. Assume that (1), , , and the conditions of hold. If , is a solution of (1) with initial data according to Definition 2 on the interval ; then for any ,

Proof. For any , we will show that (a.s.) as for any , so that the local solution becomes a global solution where is a stopping time which is defined in Section 4. Similar to [28], by the Theorem 7, for , is the local solution of (1), so the following energy equation holds:

Using Hölder’s inequality, Young’s inequality, and embedding theorem, we have

where is the embedding constant, and is a constant depending on , consequently; we have

Since , we distinguish two cases.

(1)Either so we choose so small that .(2)Or ; in this case, we have .

Hence, in either case, we have

Using the conditions of , we obtain

which implies that

Consequently we have

Taking the expectation of (114), we get

The Gronwall’s inequality implies that

On the other hand, we have

where denotes the indicator function. In view of (116) and (117), we get

The Borel-Cantelli lemma implies that for any . This shows that is the global solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by NSF of China 11272277 and 11226188 and FRF for the Central Universities of China 2013ZZGH027.

References

A. Haraux and E. Zuazua, “Decay estimates for some semilinear damped hyperbolic problems,” Archive for Rational Mechanics and Analysis, vol. 150, no. 2, pp. 191–206, 1988.

M. Kopáčková, “Remarks on bounded solutions of a semilinear dissipative hyperbolic equation,” Commentationes Mathematicae Universitatis Carolinae, vol. 30, no. 4, pp. 713–719, 1989.

J. M. Ball, “Remarks on blow-up and nonexistence theorems for nonlinear evolution equations,” The Quarterly Journal of Mathematics, vol. 28, no. 112, pp. 473–486, 1977.

V. K. Kalantarov and O. A. Ladyzhenskaya, “The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types,” Journal of Soviet Mathematics, vol. 10, no. 1, pp. 53–70, 1978.

H. A. Levine, “Some additional remarks on the nonexistence of global solutions to nonlinear wave equations,” SIAM Journal on Mathematical Analysis, vol. 5, pp. 138–146, 1974.

H. A. Levine, S. R. Park, and J. Serrin, “Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation,” Journal of Mathematical Analysis and Applications, vol. 228, no. 1, pp. 181–205, 1998.

V. Georgiev and G. Todorova, “Existence of a solution of the wave equation with nonlinear damping and source terms,” Journal of Differential Equations, vol. 109, no. 2, pp. 295–308, 1994.

C. O. Alves, M. M. Cavalcanti, V. N. D. Cavalcanti, M. A. Rammaha, and D. Toundykov, “On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms,” Discrete and Continuous Dynamical Systems. Series S, vol. 2, no. 3, pp. 583–608, 2009.

M. A. Rammaha, “The influence of damping and source terms on solutions of nonlinear wave equations,” Boletim da Sociedade Paranaense de Matemática, vol. 25, no. 1-2, pp. 77–90, 2007.

M. Kafini and S. A. Messaoudi, “A blow-up result in a Cauchy viscoelastic problem,” Applied Mathematics Letters, vol. 21, no. 6, pp. 549–553, 2008.

S. A. Messaoudi, “Blow up and global existence in a nonlinear viscoelastic wave equation,” Mathematische Nachrichten, vol. 260, pp. 58–66, 2003.

S. A. Messaoudi, “Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation,” Journal of Mathematical Analysis and Applications, vol. 320, no. 2, pp. 902–915, 2006.

H. Song and C. Zhong, “Blow-up of solutions of a nonlinear viscoelastic wave equation,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 3877–3883, 2010.

R. Ikehata, “Some remarks on the wave equations with nonlinear damping and source terms,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 27, no. 10, pp. 1165–1175, 1996.

M. Aassila, M. M. Cavalcanti, and V. N. D. Cavalcanti, “Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term,” Calculus of Variations and Partial Differential Equations, vol. 15, no. 2, pp. 155–180, 2002.

M. M. Cavalcanti, V. N. D. Cavalcanti, J. S. P. Filho, and J. A. Soriano, “Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping,” Differential and Integral Equations, vol. 14, no. 1, pp. 85–116, 2001.

M. M. Cavalcanti, V. N. D. Cavalcanti, and P. Martinez, “Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term,” Journal of Differential Equations, vol. 203, no. 1, pp. 119–158, 2004.

M. M. Cavalcanti, V. N. D. Cavalcanti, and F. A. F. Nascimento, “Asymptotic stability of the wave equation on compact manifolds and locally distributed viscoelastic dissipation,” Proceedings of the American Mathematical Society, vol. 141, no. 9, pp. 3183–3193, 2013.

P.-L. Chow, Stochastic Partial Differential Equations, Chapman & Hall/CRC, NewYork, 2007.

S. L. Cheng, Y. T. Guo, and Y. B. Tang, “Stochastic nonlinear thermoelastic system coupled sine-Gordon equation driven by jump noise,” Abstract and Applied Analysis, vol. 2014, Article ID 403528, 12 pages, 2014.

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, UK, 1996.

G. Wang and Y. B. Tang, “Fractal dimension of a random invariant set and applications,” Journal of Applied Mathematics, vol. 2013, Article ID 415764, 5 pages, 2013.

G. Wang and Y. B. Tang, “-random attractors for stochastic reaction-diffusion equation on unbounded domains,” Abstract and Applied Analysis, vol. 2013, Article ID 279509, 23 pages, 2013.

G. Wang and Y. B. Tang, “Random attractors for stochastic reaction-diffusion equations with multiplicative noise in ,” Mathematische Nachrichten, 2014.

M. Wang and Y. B. Tang, “Attractors in and for reaction diffusion equations on unbounded domains,” Communications on Pure and Applied Analysis, vol. 12, no. 2, pp. 1111–1121, 2013.

E. Wu and Y. B. Tang, “Random perturbations of reaction-diffusion waves in biology,” Wave Motion, vol. 49, no. 7, pp. 632–637, 2012.

L. Bo, D. Tang, and Y. Wang, “Explosive solutions of stochastic wave equations with damping on ,” Journal of Differential Equations, vol. 244, no. 1, pp. 170–187, 2008.

H. Gao, F. Liang, and B. Guo, “Stochastic wave equations with nonlinear damping and source terms,” Infinite Dimensional Analysis, Quantum Probability and Related Topics, vol. 16, no. 2, Article ID 1350013, 29 pages, 2013.

T. Wei and Y. Jiang, “Stochastic wave equations with memory,” Chinese Annals of Mathematics B, vol. 31, no. 3, pp. 329–342, 2010.

F. Liang and H. Gao, “Explosive solutions of stochastic viscoelastic wave equations with damping,” Reviews in Mathematical Physics, vol. 23, no. 8, pp. 883–902, 2011.

J. U. Kim, “On the stochastic wave equation with nonlinear damping,” Applied Mathematics and Optimization, vol. 58, no. 1, pp. 29–67, 2008.

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, NY, USA, 1983.

J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, France, 1969.